## The Quickhull algorithm for convex hulls (1996)

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Venue: | ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE |

Citations: | 475 - 0 self |

### BibTeX

@ARTICLE{Barber96thequickhull,

author = {C. Bradford Barber and David P. Dobkin and Hannu Huhdanpaa},

title = {The Quickhull algorithm for convex hulls},

journal = {ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE},

year = {1996},

volume = {22},

number = {4},

pages = {469--483}

}

### Years of Citing Articles

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### Abstract

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

### Citations

1804 |
Computational Geometry: An Introduction
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(Show Context)
Citation Context ... ridges of the lower convex hull is the Delaunay triangulation of the original points [9]. The intersection of halfspaces about the origin is equivalent to the convex hull of the points in dual space =-=[39]-=-. To determine the intersection of halfspaces: locate an interior point by linear programming [43], translate the interior point to the origin, transform halfspaces into points by dividing offsets int... |

582 | Voronoi diagrams – A survey of a fundamental geometric data structure
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(Show Context)
Citation Context ...ile searching, cluster analysis, collision detection, crystallography, metallurgy, urban planning, cartography, image processing, numerical integration, statistics, sphere packing, and point location =-=[2]-=-. We represent a convex hull with a set of facets and a set of adjacency lists giving the neighbors and vertices for each facet. The boundary elements of a facet are called ridges. Each ridge signifie... |

422 | Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator - Shewchuk - 1996 |

393 | New applications of random sampling in computational geometry
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- 1987
(Show Context)
Citation Context ... visible facets, thus forming the convex hull of the new point and the previously processed points. The original randomized incremental algorithm upon which we build was proposed by Clarkson and Shor =-=[16]-=-. They work in the dual space of halfspace intersections. Their algorithm adds a halfspace by intersecting it with the polytope of the previous intersections. They randomly select a halfspace to add t... |

285 |
Computational Geometry: An Introduction Through Randomized Algorithms
- Mulmuley
- 1993
(Show Context)
Citation Context ...the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [12] [15] [21] [28] [30] =-=[37]-=-. Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster algorithm for distribut... |

173 | Voronoi diagrams—a survey of a fundamental geometric data structure - Aurenhammer - 1991 |

161 |
Randomized Incremental Constructions of Delaunay and Voronoi Diagrams
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- 1992
(Show Context)
Citation Context ...ection of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [12] [15] [21] =-=[28]-=- [30] [37]. Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster algorithm for... |

157 |
Incremental Topological Flipping Works for Regular Triangulations
- Edelsbrunner, Shah
- 1996
(Show Context)
Citation Context ...ntersection of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [12] [15] =-=[21]-=- [28] [30] [37]. Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster algorith... |

101 |
The ultimate planar convex hull algorithm
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- 1986
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Citation Context ...size can be much smaller than the worst case size. In R 2 , Kirkpatrick and Seidel found an optimal output-sensitive algorithm for convex hull that runs in O(n log h) time, where h is the output size =-=[32]-=-. Clarkson & Shor give a 3-d convex hull algorithm with optimal outputsensitive expected time [16]; it was derandomized by Chazelle and Matousek [12]. In higher dimensions, the best output-sensitive a... |

92 | Four results on randomized incremental constructions
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- 1993
(Show Context)
Citation Context ... of intersection of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [12] =-=[15]-=- [21] [28] [30] [37]. Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster alg... |

85 | How good are convex hull algorithms
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- 1997
(Show Context)
Citation Context ...rliest incremental method for computing the convex hull. It is an excellent choice in high dimensions when the number of facets is much smaller than the maximum number of facets for r vertices (f r ) =-=[3]-=- [25]. 2. The Quickhull Algorithm We assume that the input points are in general position (i.e., no set of d + 1 points define a (d \Gamma 1)-flat), so that their convex hull is a simplicial complex [... |

79 | Prodon: Double description method revisited - Fukuda, A - 1996 |

75 |
Automated spectral unmixing of AVIRIS data using convex geometry concepts
- Boardman
- 1993
(Show Context)
Citation Context ...set that contains the points. The convex hull is a fundamental construction for mathematics and computational geometry. For example, Boardman uses the convex hull in his analysis of spectrometry data =-=[6]-=- and Weeks uses the convex hull to determine the canonical triangulation of cusped hyperbolic 3-manifolds [44]. Other problems can be reduced to the convex hull -- for example, halfspace intersection,... |

73 |
Construction of Three-Dimensional Delaunay Triangulations Using Local Transformations
- Joe
- 1991
(Show Context)
Citation Context ...n of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [12] [15] [21] [28] =-=[30]-=- [37]. Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster algorithm for dist... |

71 |
Constructing higher-dimensional convex hulls at logarithmic cost per face
- Seidel
- 1986
(Show Context)
Citation Context ...ted time [16]; it was derandomized by Chazelle and Matousek [12]. In higher dimensions, the best output-sensitive algorithm is Seidel's shelling algorithm at O(n 2 + h log n) when h = \Omega\Gamma n) =-=[40]-=-, and gift-wrapping at O(nh) otherwise [11]. The Double-Description Method is the dual of the Beneath-Beyond Algorithm [36]. It is the earliest incremental method for computing the convex hull. It is ... |

69 |
Voronoi diagrams from convex hulls
- BROWN
- 1979
(Show Context)
Citation Context ...launay triangulation of a set of points: lift the points to a paraboloid and compute their convex hull. The set of ridges of the lower convex hull is the Delaunay triangulation of the original points =-=[9]-=-. The intersection of halfspaces about the origin is equivalent to the convex hull of the points in dual space [39]. To determine the intersection of halfspaces: locate an interior point by linear pro... |

66 |
Thrall: The double description method
- Motzkin, Raiffa, et al.
- 1953
(Show Context)
Citation Context ... is Seidel's shelling algorithm at O(n 2 + h log n) when h = \Omega\Gamma n) [40], and gift-wrapping at O(nh) otherwise [11]. The Double-Description Method is the dual of the Beneath-Beyond Algorithm =-=[36]-=-. It is the earliest incremental method for computing the convex hull. It is an excellent choice in high dimensions when the number of facets is much smaller than the maximum number of facets for r ve... |

63 |
An algorithm for convex polytopes
- Chand, Kapur
- 1970
(Show Context)
Citation Context ...lle and Matousek [12]. In higher dimensions, the best output-sensitive algorithm is Seidel's shelling algorithm at O(n 2 + h log n) when h = \Omega\Gamma n) [40], and gift-wrapping at O(nh) otherwise =-=[11]-=-. The Double-Description Method is the dual of the Beneath-Beyond Algorithm [36]. It is the earliest incremental method for computing the convex hull. It is an excellent choice in high dimensions when... |

53 | Safe and effective determinant evaluation
- Clarkson
- 1992
(Show Context)
Citation Context ...olume. Clarkson's hull implementation of the randomized incremental algorithm restricts the input precision to about fifteen decimal digits. The implementation computes the exact sign of determinants =-=[13]-=-. It is a practical solution for precise convex hulls and Delaunay triangulations [14]. We timed hull, hullio[19] (a precursor of hull without exact arithmetic), triangle[41] (a two dimensional Delaun... |

46 |
Stable maintenance of point set triangulations in two dimensions
- Fortune
- 1989
(Show Context)
Citation Context ...oplanar point may be far above a new facet. If this occurs, Quickhull generates a warning and reports a wide facet. In R 2 , there are several robust convex hull and Delaunay triangulation algorithms =-=[23]-=- [29] [34]. In R 3 , Sugihara and Dey et. al. produce a topologically robust convex hull and Delaunay triangulation [18] [42]. Their algorithms are a variation of Beneath-Beyond with steps to prevent ... |

36 |
Measures of symmetry for convex sets
- Grünbaum
- 1963
(Show Context)
Citation Context ...ith interior points. An incremental algorithm for the convex hull repeatedly adds a point to the convex hull of the previously processed points. Of particular interest is the Beneath-Beyond Algorithm =-=[27]-=- [31] [39]. A new point is processed in three steps. First, locate the visible facets for the point The boundary of the visible facets is the set of horizon ridges for the point. A facet is visible if... |

30 |
A new convex hull algorithm for planar sets
- Eddy
- 1977
(Show Context)
Citation Context ...n unprocessed point is in exactly one outside set. Our variation is to process the furthest point of an outside set instead of a random point. In R 2 , this is the well-known Quickhull Algorithm [10] =-=[20]-=- [22] [26]. Other variations of the Clarkson and Shor algorithm do not maintain conflict graphs or outside sets. Instead, they retain old facets of the convex hull with links to the new facets that re... |

29 | Constructing strongly convex hulls using exact or rounded arithmetic
- Li, Milenkovic
- 1992
(Show Context)
Citation Context ...int may be far above a new facet. If this occurs, Quickhull generates a warning and reports a wide facet. In R 2 , there are several robust convex hull and Delaunay triangulation algorithms [23] [29] =-=[34]-=-. In R 3 , Sugihara and Dey et. al. produce a topologically robust convex hull and Delaunay triangulation [18] [42]. Their algorithms are a variation of Beneath-Beyond with steps to prevent topologica... |

27 |
Derandomizing an output-sensitive convex hull algorithm in three dimensions
- Chazelle, Matousek
- 1995
(Show Context)
Citation Context ...point of intersection of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance =-=[12]-=- [15] [21] [28] [30] [37]. Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faste... |

26 |
Triangle: A two-dimensional quality mesh generator and delaunay triangulator. Website
- Shewchuk
- 2006
(Show Context)
Citation Context ...xact sign of determinants [13]. It is a practical solution for precise convex hulls and Delaunay triangulations [14]. We timed hull, hullio[19] (a precursor of hull without exact arithmetic), triangle=-=[41]-=- (a two dimensional Delaunay triangulation program with exact arithmetic), and our implementation of Quickhull (qhull 2.2) on a Silicon Graphics 100 MHz R4000. These are fastest implementations known ... |

25 | On the randomized construction of the Delaunay tree, Theoret - Boissonnat, Teillaud - 1993 |

21 |
Constructing strongly convex approximate hulls with inaccurate primitives
- Guibas, Salesin, et al.
- 1993
(Show Context)
Citation Context ...ar point may be far above a new facet. If this occurs, Quickhull generates a warning and reports a wide facet. In R 2 , there are several robust convex hull and Delaunay triangulation algorithms [23] =-=[29]-=- [34]. In R 3 , Sugihara and Dey et. al. produce a topologically robust convex hull and Delaunay triangulation [18] [42]. Their algorithms are a variation of Beneath-Beyond with steps to prevent topol... |

20 | On the randomized construction of the Delaunay tree
- Boissonnat, Teillaud
- 1993
(Show Context)
Citation Context ...acets tested for a point. Quickhull may test the same sequence during successive partitions of the point into outside sets. Edelsbrunner and Shah [21], Joe [30], and Boissonnat and Devillers-Teillaud =-=[7]-=- use a similar method for Delaunay triangulations. They express their algorithm in terms of triangulations and the in-sphere test. By the correspondence between Delaunay triangulation and convex hull,... |

15 |
Convex hull of a finite set of points in two dimensions
- Bykat
- 1978
(Show Context)
Citation Context ...hm, an unprocessed point is in exactly one outside set. Our variation is to process the furthest point of an outside set instead of a random point. In R 2 , this is the well-known Quickhull Algorithm =-=[10]-=- [20] [22] [26]. Other variations of the Clarkson and Shor algorithm do not maintain conflict graphs or outside sets. Instead, they retain old facets of the convex hull with links to the new facets th... |

15 |
Convex hulls and isometries of cusped hyperbolic 3-manifolds
- Weeks
- 1993
(Show Context)
Citation Context ... geometry. For example, Boardman uses the convex hull in his analysis of spectrometry data [6] and Weeks uses the convex hull to determine the canonical triangulation of cusped hyperbolic 3-manifolds =-=[44]-=-. Other problems can be reduced to the convex hull -- for example, halfspace intersection, Delaunay 116 Fayerweather Street, Cambridge, MA 02138, bradb@geom.umn.edu. This research was supported in par... |

14 |
Learning in Navigation: Goal finding in graphs
- Cucka, Netanyahu, et al.
- 1996
(Show Context)
Citation Context ...rol, geographic information systems, neighbors of the origin in the R 8 lattice, stress analysis, stability of robot grasps [5], spectrometry [6], constrained control allocation [8], robot navigation =-=[17]-=-, micromagnetic modeling [38], and invariant sets of delta-sigma modulators [45]. Acknowledgments: A special thanks to Albert Marden and Victor Milenkovic for providing excellent environments for comp... |

13 |
Determination and evaluation of support structures in layered manufacturing
- Allen, Dutta
- 1995
(Show Context)
Citation Context ...tatistics. The program can be called from within an application. Over the last two years, 3000 copies of qhull were retrieved via ftp. It has been used for support structures in layered manufacturing =-=[1]-=-, classification of molecules by their biological activity, vibration control, geographic information systems, neighbors of the origin in the R 8 lattice, stress analysis, stability of robot grasps [5... |

11 | Computational geometry
- Fortune
- 1993
(Show Context)
Citation Context ... hull, or all facets of the current hull are above the point and the point is discarded. These algorithms have been implemented. In practice, their running times are competitive with other algorithms =-=[24]-=-. We can compare Quickhull with the randomized incremental algorithms by changing the selection step of Quickhull. If Quickhull selects a random point instead of a furthest point, it is a randomized i... |

10 |
Delauny triangulations in three dimensions with finite precision arithmetic
- Dey, Sugihara, et al.
- 1992
(Show Context)
Citation Context ... 2 , there are several robust convex hull and Delaunay triangulation algorithms [23] [29] [34]. In R 3 , Sugihara and Dey et. al. produce a topologically robust convex hull and Delaunay triangulation =-=[18]-=- [42]. Their algorithms are a variation of Beneath-Beyond with steps to prevent topological anomalies such as in Figure 3. The output may contain unbounded geometric faults. There are several implemen... |

9 | cdd/cdd+ Reference Manual
- Fukuda
(Show Context)
Citation Context ...st incremental method for computing the convex hull. It is an excellent choice in high dimensions when the number of facets is much smaller than the maximum number of facets for r vertices (f r ) [3] =-=[25]-=-. 2. The Quickhull Algorithm We assume that the input points are in general position (i.e., no set of d + 1 points define a (d \Gamma 1)-flat), so that their convex hull is a simplicial complex [39]. ... |

8 |
Private communication
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- 1991
(Show Context)
Citation Context ...rocessed point is in exactly one outside set. Our variation is to process the furthest point of an outside set instead of a random point. In R 2 , this is the well-known Quickhull Algorithm [10] [20] =-=[22]-=- [26]. Other variations of the Clarkson and Shor algorithm do not maintain conflict graphs or outside sets. Instead, they retain old facets of the convex hull with links to the new facets that replace... |

8 |
Constructing the convex hull of a set of points in the plane
- Green, Silverman
- 1979
(Show Context)
Citation Context ...sed point is in exactly one outside set. Our variation is to process the furthest point of an outside set instead of a random point. In R 2 , this is the well-known Quickhull Algorithm [10] [20] [22] =-=[26]-=-. Other variations of the Clarkson and Shor algorithm do not maintain conflict graphs or outside sets. Instead, they retain old facets of the convex hull with links to the new facets that replaced the... |

8 |
Convex polytopes and linear programming
- Klee
- 1966
(Show Context)
Citation Context ...s two balance conditions. Let d be the dimension, n be the number of input points, r be the number of processed points, and f r be the maximum number of facets of r vertices (f r = O(r bd=2c =bd=2c!) =-=[33]-=-). Definition . An execution of Quickhull is balanced if ffl the average number of new facets for the j-th processed point is df j =j ffl the average number of partitioned points for the j-th processe... |

6 |
W.C.: Closed-form solutions to constrained control allocation problem
- Bordignon, Durham
- 1995
(Show Context)
Citation Context ...tivity, vibration control, geographic information systems, neighbors of the origin in the R 8 lattice, stress analysis, stability of robot grasps [5], spectrometry [6], constrained control allocation =-=[8]-=-, robot navigation [17], micromagnetic modeling [38], and invariant sets of delta-sigma modulators [45]. Acknowledgments: A special thanks to Albert Marden and Victor Milenkovic for providing excellen... |

3 |
Convex hull algorithms in higher dimensions
- Kallay
- 1981
(Show Context)
Citation Context ...nterior points. An incremental algorithm for the convex hull repeatedly adds a point to the convex hull of the previously processed points. Of particular interest is the Beneath-Beyond Algorithm [27] =-=[31]-=- [39]. A new point is processed in three steps. First, locate the visible facets for the point The boundary of the visible facets is the set of horizon ridges for the point. A facet is visible if the ... |

3 |
Invariant sets for general second-order low-pass delta-sigma modulators with dc inputs
- Zhang, Goodson, et al.
- 1994
(Show Context)
Citation Context ... stress analysis, stability of robot grasps [5], spectrometry [6], constrained control allocation [8], robot navigation [17], micromagnetic modeling [38], and invariant sets of delta-sigma modulators =-=[45]-=-. Acknowledgments: A special thanks to Albert Marden and Victor Milenkovic for providing excellent environments for completing this work. The referees' comments greatly improved the presentation and c... |

2 | A program for convex hulls - CLARKSON - 1995 |

1 |
Personal communications regarding qhull
- Belsis, Grundmann, et al.
- 1995
(Show Context)
Citation Context ...1], classification of molecules by their biological activity, vibration control, geographic information systems, neighbors of the origin in the R 8 lattice, stress analysis, stability of robot grasps =-=[5]-=-, spectrometry [6], constrained control allocation [8], robot navigation [17], micromagnetic modeling [38], and invariant sets of delta-sigma modulators [45]. Acknowledgments: A special thanks to Albe... |

1 |
A program for convex hulls. http://netlib.att.com/netlib/ voronoi/hull.html
- Clarkson
- 1995
(Show Context)
Citation Context ...s the input precision to about fifteen decimal digits. The implementation computes the exact sign of determinants [13]. It is a practical solution for precise convex hulls and Delaunay triangulations =-=[14]-=-. We timed hull, hullio[19] (a precursor of hull without exact arithmetic), triangle[41] (a two dimensional Delaunay triangulation program with exact arithmetic), and our implementation of Quickhull (... |

1 | Irregular grain structure in micromagnetic simulation
- Porter, Glavinas, et al.
- 1996
(Show Context)
Citation Context ...ystems, neighbors of the origin in the R 8 lattice, stress analysis, stability of robot grasps [5], spectrometry [6], constrained control allocation [8], robot navigation [17], micromagnetic modeling =-=[38]-=-, and invariant sets of delta-sigma modulators [45]. Acknowledgments: A special thanks to Albert Marden and Victor Milenkovic for providing excellent environments for completing this work. The referee... |

1 |
Topologically consistent algorithms related to convex polyhedra
- Sugihara
- 1992
(Show Context)
Citation Context ...there are several robust convex hull and Delaunay triangulation algorithms [23] [29] [34]. In R 3 , Sugihara and Dey et. al. produce a topologically robust convex hull and Delaunay triangulation [18] =-=[42]-=-. Their algorithms are a variation of Beneath-Beyond with steps to prevent topological anomalies such as in Figure 3. The output may contain unbounded geometric faults. There are several implementatio... |

1 | The Quickhull Algorithm for Convex Hulls • 481 - ALLEN, DUTTA - 1995 |

1 | The Quickhull Algorithm for Convex Hulls • 483 - MOTZKIN, RAIFFA, et al. - 1953 |

1 | Convex hulls and isometrics of cusped hyperbolic 3-manifolds - WEEKS - 1991 |

1 | Determination& evaluationof support structures in layered manufacturing - Allen, Dutta - 1995 |